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Could Nature, LHC prefer N=2 supersymmetry?

An even more ambitious, more stringy type of supersymmetry could actually be more compatible with the LHC data

A year ago or so, we have entered the serious LHC era in which many eyes of high-energy physicists have been refocused from formal, top-down theory to experiments and phenomenology, i.e. to the thinking about the signs of new physics we may soon see.

While the speed with which the LHC may uncover new physics has surely been disappointing for many victims of a wishful thinking, it can't be excluded that some signs of new physics will emerge in a few months or years. Aside from the Higgs boson(s), supersymmetry remains the most likely scheme that could appear as the first discovery.

Minimal and extended supersymmetry

Supersymmetry is a new, very abstract form of a symmetry. It is mathematically analogous to ordinary symmetries such as the rotational symmetry \(SO(3)\).

The rotational symmetry forms something known as a Lie (continuous) group and a very efficient way to study and classify such possible symmetries is to look at the infinitesimal transformations (transformations by infinitely small angles). Those may be expressed as tiny variations of the identity matrix, \({\bf 1}+iM\), where \(M\) is typically an infinitesimal Hermitian matrix (I needed the \(i\) factor for the sum to be unitary).

The information about the "shape" of the rotational symmetry group or another group is encoded in the linear space of possible matrices \(M\). This space is known as the Lie algebra and contains an operation \([M,N]\), the commutator, which knows everything about the multiplication rules of the original group as well as its "curved structure". The commutator may be viewed as an abstract operation but if you represent the generators by genuine particular matrices, it reduces to \(MN-NM\), indeed.

Supersymmetry is one of the "generalized forms of a Lie group" which is also determined by its Lie algebra of infinitesimal transformations. However, it must be a "superalgebra" which means that the objects \(M,N\) are no longer ordinary operators that have real eigenvalues we may imagine. Instead, the generators of a supersymmetry algebra are operators such as \(Q_\alpha\) which are "anticommuting" or "Grassmann-odd". These adjectives are also morally equivalent to "fermionic" which contrasts with the adjective "bosonic" or "commuting" or "Grassmann-even" which may be used for the original generators \(M,N\) of an ordinary Lie algebra.

Grassmann-odd, fermionic numbers don't commute with each other. Instead, in the simplest case, they anticommute,
\[ \{\theta_i,\theta_j\}\equiv \theta_i \theta_j + \theta_j\theta_i = 0. \] If you exchange the ordering in the multiplication, you change the sign of the product. This is analogous e.g. to a cross product of two vectors, \(\vec a\times \vec b = -\vec b \times \vec a\). However, the multiplication of the Grassmann-odd numbers (and even operators) is still associative, unlike the cross product, and you can't ever represent Grasmann-odd numbers by ordinary real numbers. Instead, they're numbers that can't be "enumerated". They don't belong to any well-defined set. You may only get "ordinary real or complex numbers" if you multiply an even number of Grassmann-odd numbers.

Based on this description, you could guess that the Grassmann-odd numbers are "unreal" in a similar way as a wave function. The absolute value of a wave function must be first squared to get something with a clear measurable interpretation, namely a probability. The same thing holds for Grassmann-odd numbers. They would really make no sense (or would be identically equal to zero) in any classical theory. But in a quantum theory, they're as sensible as the ordinary numbers as a description of wave functions and operators.

Much like bosons and fermions are equally natural or legitimate, Grassmann-even and Grassmann-odd numbers are equally legitimate as well. This conclusion would almost certainly be surprising for someone who hasn't studied quantum field theory in detail: but once he studies it, he understands it's an established fact. When you exchange two identical particles (or two "distant" operators of the same kind), you can't change the physics. So the probability distributions shouldn't change; but if the wave function gets multiplied by \(\pm 1\), it's still OK because even \(-1\) squares to one which is good because you get back to the original state if you do these permutations twice; and because the probabilities are unchanged because probabilities are squared amplitudes.

It's time to say what the supersymmetry algebra is. It has the generators \(Q_\alpha\) which must transform as spacetime spinors. They commute with the Hamiltonian so they're "conserved charges". It's remarkable that one may have conserved charges carrying spin 1/2 at all; it is one of the counterexample to the Coleman-Mandula theorem (if you formulated this theorem in a sloppy way and omitted some of its assumptions). If you know what spinors are, and I am not going to explain it here, you know that the dimension of a spinor representation grows as a power of two where the exponent is roughly one half of the spacetime dimension:
\[ D({\rm spinor}) = 2^{D({\rm spacetime})/2} \] Well, you should really subtract \(1\) from the spacetime dimension first, and then take the integer part of one-half of the result, to get the dimension of the smallest irreducible spinor representation. Let's not go into these technicalities. The fact that \(Q_\alpha\) are spinors determines their commutator with the Lorentz generators. They commute with the energy-momentum, as I suggested previously. And the most important new commutator – well, this one and only this one is an anticommutator – says
\[ \{Q_\alpha,Q_\beta\} = p_\mu \gamma^\mu_{\alpha\beta} + {\rm central \,\,charges} \] It's still a bit schematic – I am not distinguishing dotted and undotted spinor indices etc. – but it contains the right flavor of the truth. (The anticommutator given by the curly bracket is just like a commutator but with a plus sign.) In words, the supersymmetry generators anticommute to energy-momentum, i.e. ordinary spacetime translations: the gamma matrices are just coefficients needed for the equation to be nicely Lorentz-covariant (i.e. for it to respect the pairing of indices etc.). There may also be additional similar terms on the right hand side known as "central charges" – various conserved winding numbers of strings and wrapping numbers of branes are good examples in string/M-theory.

Because the number of the components of a spinor grows quickly with the spacetime dimension and because each generator of a symmetry constrains your theory – and supersymmetry is no exception – you quickly get an overconstrained theory. So the spacetime dimension shouldn't be too high. It turns out that the maximum spacetime dimension in which you may get a non-trivial, interacting theory with \(D\) large spacetime dimensions which also contains massless particles and no negative probabilities is \(D=11\). The corresponding theory has 32 "real supercharges" – generators of the supersymmetry algebra which are rewritten as a collection of independent Hermitian Grassmann-odd operators in this counting.

The effective supersymmetric field theory in \(D=11\) inevitably contains gravity because the 32 supercharges are able to change the spin of particles by a few units and you inevitably get to \(j=2\) of a graviton, too. And it is called the eleven-dimensional supergravity. It is arguably the most symmetric and most beautiful or "the simplest" extension of Einstein's general theory of relativity. As a quantum field theory, it is probably non-renormalizable perturbatively, with new counterterms arising at 7 loops, and even if they cancel to all orders, it is surely inconsistent at the non-perturbative level. The unique short-distance completion (a fully consistent theory, at all distances, that reduces to the other one at long distances) of this theory is nothing else than the eleven-dimensional M-theory, a "higher-dimensional sibling" of the ten-dimensional superstring vacua.

Going to lower dimensions

I could describe how the possible number of supersymmetry generators depends on dimensions below 11 but let's jump directly to \(D=4\) because many readers still incorrectly think that our spacetime only has 4 dimensions. The \(D=4\) Weyl/chiral spinor has two complex components which carry the information of 4 real components. Alternatively, you may describe them as 4 real components of a Majorana/real spinor. The Majorana and Weyl conditions can't be imposed simultaneously in \(D=4\). Instead, you only have either Weyl spinors or the Majorana spinors as the minimal ones, and they're pretty much the same thing when it comes to the information they carry (although we typically manipulate with them differently). The Dirac spinor is made out of two Weyl (or two Majorana) spinors.

If you compactify eleven-dimensional M-theory on a 7-dimensional torus, the original 32 real supercharges in 11 dimensions are divided into 8 groups of 4 supercharges, i.e. 8 Weyl spinors. One Weyl spinor of supercharges is the minimum amount of supersymmetry you may have in \(D=4\), the so-called minimal or \({\mathcal N}=1\) supersymmetry. On the other hand, the 32 supercharges inherited from the maximally supersymmetric theory, M-theory, produce the maximal or \({\mathcal N} = 8 \) supergravity. Numbers in between are possible, especially the important cases \({\mathcal N}=2 \) and \( {\mathcal N}=4\) but some values that are not powers of two are possible, too (but much less attractive for the researchers, at least so far).

I can't go into details but the more supersymmetries you get, the more accurately you may calculate various physical predictions. The "simplest compactification" in M-theory or string theory have a large number of supercharges, they correspond to simple compactification manifolds (simple shape of hidden dimensions), they are relatively unique, and they are pretty well-understood.

On the other hand, vacua with a small amount of supersymmetry are numerous, unconstrained, they form large portions of the "landscape", and supersymmetry is only slightly helpful if you want to easily calculate predictions: many terms in all these predictions must be calculated, anyway, and sometimes the exact results are not known. Non-supersymmetric vacua are the messiest ones, of course. They're really not under control, they're potentially unstable, they may fail to exist at all, and they may also form an even larger landscape than the supersymmetric ones (or they may have a miraculous, not quite understood reason why they are or why it is unique). We have many guesses but we don't know for sure.

How many supersymmetries are in the real world?

Ten-dimensional string vacua have either 16 or 32 real supercharges, either the full 11-dimensional amount or one half of it. Because the 10-dimensional spinor is smaller than the 11-dimensional one – there exists a Majorana/Weyl spinor in \(D=10\) that is both real and chiral, and therefore allows one to reduce the dimension from 32 real to 16 real components – we are able to get below 32.

Heterotic string theory, the first 10-dimensional stringy starting point that was shown to be capable of explaining all types of particles and forces we observe in Nature, has 16 supercharges to start with. You may say that it's only 1/2 of the maximal number because the supercharges only arise from the left-moving excitations moving along strings, while all the supersymmetries linked to the right-moving excitations are eliminated by replacing the right-moving excitations by the old 26-dimensional bosonic string theory.

When you compactify a theory on a Calabi-Yau three-fold, you reduce the number of real supercharges to 1/4 of the original number. So you may see that by compactifying a heterotic string theory on a Calabi-Yau three-fold, you get a nice 4-dimensional theory with the minimal \({\mathcal N} = 1\) supersymmetry. Even the minimal supersymmetry has some big advantages over no supersymmetry but the next "extended" i.e. \({\mathcal N}>1\) supersymmetr, namely \({\mathcal N}=2\) supersymmetry, already seems "too much of a good thing".

Why? If you extend the known list of particles to all of their supersymmetric cousins, you will be able to derive that the resulting theory has too many particle species, too few independent interactions, and doesn't allow many things such as the CP-violation and the electroweak "chiral" couplings of the gauge field to the fermions, either. For years, everyone would assume that only the minimal, \({\mathcal N}=1\) supersymmetry, may be preserved in Nature down to energies that are approximately accessible to our accelerators. (At very high energies, probably inaccessible experimentally, the maximum \( {\mathcal N}=8\) SUSY still exists morally in some sense, namely locally on the manifold where it looks like a flat 10-dimensional type II string theory or 11-dimensional M-theory.)

A fun quotation. At TASI 99, Paul Aspinwall was one of the lecturers. He gave great lectures about the \({\mathcal N}=2\) theories which are interesting for many mathematical reasons; the Seiberg-Witten analysis of magnetic monopoles and monodromies in gauge theories belongs here. Paul Aspinwall would say that \({\mathcal N}=2\) is the most beautiful, naturally balanced choice interpolating between the too high supersymmetries which are too constraining and make the theories too trivial, and the minimal or no supersymmetry which is too unconstrained and messy. "The only problem is that our world doesn't seem to have an \({\mathcal N}=2\) supersymmetry, but this is not my fault."

Everyone laughed. But was he right? Finally, I am getting to the new hep-ph paper:

How Many Supersymmetries?
It was written in Toronto, Canada by Matti Heikinheimo, Moshe Kellerstein, and Veronica Sanz. They argue that \({\mathcal N}=2\) is not only possible but actually favored by the existing LHC exclusion limits.

Before the experts leave this article with the word "obvious bullshit", let me say that they only talk about the extended supersymmetry of the gauge sector. It would really lead to contradictions if you tried to extend the quarks and leptons to \({\mathcal N}=2\) multiplets.

Does it make any sense for some particles, but not others, to respect the extended supersymmetry? Well, it could make more sense than what you might think. In a braneworld scenario of string theory (and this is of course my comment, but one that you may hear from pretty much any string theorist), the gauge bosons may live on branes that preserve a higher number of supersymmetries than the branes or their intersections where the leptons and quarks live. Of course, the supersymmetry respected by the "whole theory" is just the greatest common denominator, i.e. at most \({\mathcal N}=1\) which is spontaneously broken, anyway. But the gauge fields could have more supercharges.

The non-gravitational forces we know, the electroweak force and the strong force, could actually naturally be extended to whole \({\mathcal N}=2\) multiplets. Instead of a gluon and a single gluino, you would have a gluon, two gluinos, and a bosonic scalar sgluino (or whatever the name should be). We know such a "replacement" of non-supersymmetric theories by highly (extended) supersymmetric theories (or an even more extended one) from another context: AdS/QCD. When applied to heavy ion physics and nuclear physics in general, people want to understand phenomena that seem to agree with the old non-supersymmetric QCD. However, they often use the maximally supersymmetry \({\mathcal N}=4\) gauge theory and they get a good agreement. An extended supersymmetrization of the gauge fields seems to be a rather harmless, and probably healthy, modification.

The authors of the paper show the methods how to distinguish \({\mathcal N}=1\) from \({\mathcal N}=2\) supersymmetry at the LHC. And they even find out that with the extended, more ambitious supersymmetry, the lower limits on the squark masses actually weaken! So if the gauge fields describing the three forces are described by a theory with extended supersymmetry, some of the supersymmetric particles could actually be lighter than what a naive analysis of the current LHC data seems to allow!

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